A355356 - OEIS

%I #4 Jun 30 2022 10:40:25

%S 1,0,1,3,10,28,79,216,603,1702,4933,14620,44287,136352,424858,1334162,

%T 4211572,13344072,42412667,135217722,432483522,1387929369,4469341807,

%U 14439523193,46795072968,152076428228,495460089510,1617787324674,5292984017236,17348743335252

%N G.f. A(x) satisfies: x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%C a(n) = Sum_{k=0..floor(n/2)} A355350(n-k,k) for n >= 0.

%F G.f. A(x) satisfies:

%F (1) x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%F (2) x^2*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

%e G.f.: A(x) = 1 + x^2 + 3*x^3 + 10*x^4 + 28*x^5 + 79*x^6 + 216*x^7 + 603*x^8 + 1702*x^9 + 4933*x^10 + 14620*x^11 + 44287*x^12 + ...

%e where

%e x^2 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...

%e also,

%e x^2*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...

%e where P(x) is the partition function and begins

%e P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

%o (PARI) {a(n) = my(A=[1,0,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));

%o A[#A] = polcoeff( x^2 - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A355350, A355351, A355352, A355353, A355354, A355355, A355357.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jun 29 2022